tan(78)=4.7046 approx; sec(78)=4.8097 approx, therefore sec(x)sec(78)/2=2.4049/cos(x). This is the right-hand side.
Is cot(2x)-4.7046=2.4049/cos(x)? If this is an identity, any value of x will make it true. Put x=45, cot(2x)=cot(90)=tan(0)=0. cos(45)=√2/2=0.7071 approx. But this would mean -4.7046=2.4049√2=3.4010. So the two sides are different, and this is not an identity. So we need to solve for specific value(s) of x.
Amazingly, the answer is x=4, so now we need to find out why.
See answer to other question: TITLE: Cot2x - tan 78 = ( sec x sec 78 )/2